Motion in a Straight Line

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### Introduction - **Mechanics:** Branch of physics dealing with motion. - **Kinematics:** Describes motion without considering its causes. - **Dynamics:** Describes motion considering its causes (forces). - **Particle:** An object whose size can be neglected. ### Position, Distance, Displacement - **Position ($x$):** Location of an object relative to an origin. - **Distance:** Total path length covered by an object. Always non-negative. Scalar quantity. - **Displacement ($\Delta x$):** Change in position. $\Delta x = x_f - x_i$. Can be positive, negative, or zero. Vector quantity. - Example: If an object moves from $x=2m$ to $x=5m$, $\Delta x = 3m$. If it moves from $x=5m$ to $x=1m$, $\Delta x = -4m$. ### Speed and Velocity - **Average Speed:** Total distance / Total time. Scalar quantity. - Formula: $v_{avg, speed} = \frac{\text{Total Distance}}{\Delta t}$ - **Average Velocity ($\vec{v}_{avg}$):** Total displacement / Total time. Vector quantity. - Formula: $\vec{v}_{avg} = \frac{\Delta \vec{x}}{\Delta t} = \frac{\vec{x}_f - \vec{x}_i}{t_f - t_i}$ - **Instantaneous Velocity ($\vec{v}$):** Velocity at a specific instant. - Formula: $\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{x}}{\Delta t} = \frac{d\vec{x}}{dt}$ - **Instantaneous Speed:** Magnitude of instantaneous velocity. ### Acceleration - **Average Acceleration ($\vec{a}_{avg}$):** Change in velocity / Total time. Vector quantity. - Formula: $\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}_f - \vec{v}_i}{t_f - t_i}$ - **Instantaneous Acceleration ($\vec{a}$):** Acceleration at a specific instant. - Formula: $\vec{a} = \lim_{\Delta t \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d\vec{v}}{dt} = \frac{d^2\vec{x}}{dt^2}$ - **Units:** $m/s^2$ - **Direction:** If velocity and acceleration are in the same direction, speed increases. If opposite, speed decreases. ### Kinematic Equations (Constant Acceleration) These equations apply only when acceleration ($\vec{a}$) is constant. 1. **Velocity-Time Relation:** $\vec{v} = \vec{v}_0 + \vec{a}t$ 2. **Displacement-Time Relation:** $\vec{x} = \vec{x}_0 + \vec{v}_0 t + \frac{1}{2}\vec{a}t^2$ - Often written as $\Delta x = v_0 t + \frac{1}{2}at^2$ if $\vec{x}_0 = 0$. 3. **Velocity-Displacement Relation:** $\vec{v}^2 = \vec{v}_0^2 + 2\vec{a}(\vec{x} - \vec{x}_0)$ - Often written as $v^2 = v_0^2 + 2a\Delta x$. 4. **Displacement using Average Velocity:** $\Delta x = \frac{(\vec{v}_0 + \vec{v})}{2}t$ Where: - $\vec{v}_0$ = initial velocity - $\vec{v}$ = final velocity - $\vec{a}$ = constant acceleration - $t$ = time interval - $\vec{x}_0$ = initial position - $\vec{x}$ = final position - $\Delta x$ = displacement ### Free Fall (Motion under Gravity) - **Definition:** Motion of an object solely under the influence of gravity. - **Acceleration due to gravity ($g$):** Approximately $9.8 \, m/s^2$ (downwards). - For calculations, often $g \approx 10 \, m/s^2$. - The kinematic equations can be applied by replacing $\vec{a}$ with $\vec{g}$ (or $-g$ if upward is positive). - Example: If upward is positive: - $v = v_0 - gt$ - $y = y_0 + v_0 t - \frac{1}{2}gt^2$ - $v^2 = v_0^2 - 2g(y - y_0)$ ### Graphs of Motion - **Position-Time (x-t) Graph:** - Slope = Velocity - Straight line: Constant velocity - Curved line: Changing velocity (acceleration) - **Velocity-Time (v-t) Graph:** - Slope = Acceleration - Area under curve = Displacement - Straight line (non-zero slope): Constant acceleration - Horizontal line: Constant velocity (zero acceleration) - **Acceleration-Time (a-t) Graph:** - Area under curve = Change in velocity - Horizontal line: Constant acceleration